Group decision-making algorithm with sine trigonometric r,s,t-spherical fuzzy aggregation operators and their application

r, s, t-spherical fuzzy (r, s, t-SPF) sets provide a robust framework for managing uncertainties in decision-making, surpassing other fuzzy sets in their ability to accommodate diverse uncertainties through the incorporation of flexible parameters r, s, and t. Considering these characteristics, this article explores sine trigonometric laws to enhance the applicability and theoretical foundation for r, s, t-SPF setting. Following these laws, several aggregation operators (AOs) are designed for aggregation of the r, s, t-SPF data. Meanwhile, the desired characteristics and relationships of these operators are studied under sine trigonometric functions. Furthermore, we build a group decision-making algorithm for addressing multiple attribute group decision-making (MAGDM) problems using the developed AOs. To exemplify the applicability of the proposed algorithm, we address a practical example regarding laptop selection. Finally, parameter analysis and a comprehensive comparison with existing operators are conducted to uncover the superiority and validity of the presented AOs.

1. Foundational operations: What are the fundamental operations conducted by ST aggregation tools within the r, s, t-SPFS framework, and how do these operations facilitate periodic aggregation during the process?2. Innovative aggregation operators: How do the newly introduced aggregation operators, namely ST-weighted averaging, ST-weighted geometric, ST-ordered weighted averaging, ST-ordered weighted geometric, SThybrid averaging, and ST-hybrid geometric, perform within the context of r, s, t-SPFS?What are the distinctive characteristics and exceptional scenarios associated with each operator, and how do they improve the aggregation of intricate information?

Sine trigonometric operational laws based on r,s,t-SPFSs
This section outlines new operations for r, s, t-SPFSs and examines their fundamental properties.

Definition 9
Let L = (ρ, χ, ε) be any r,s,t-SPFN.If The function sin Q operates as a sine trigonometric operator, and its resultant value is referred to as the sine trigonometric r,s,t-SPFN (ST-r,s,t-SPFN).
Proof These can be easily verified from Definition 8.
Theorem 5 For any r,s,t-SPFN L and real number ⋓ > 0 , ⋓ sin L ≥ (sin Proof Based on the previous results, one can easily get the proof.

Aggregation operators based on Sine trigonometric r,s,t-SPFNs
We construct the subsequent weighted and geometric AOs relying on STOLs of r,s,t-SPFNs.Let σ be the collection of r,s,t-SPFNs L i = (ρ i , χ i , ε i ) .After that, we label it be the group of "n" r,s,t-SPFNs, for (i = 1, 2, ..., n) .A ST-r,s,t-SPFWA is a mapping: σ n → σ and defined as; where ϕ = (ϕ 1 , ϕ 2 , ..., ϕ n ) T is the weight vector of sin L i with ϕ 1 > 0 and Theorem 6 Let L i = (ρ i , χ i , ε i ) be the group of "n" r,s,t-SPFNs.Then, the aggregated value obtained by ST-r,s,t- SPFWA operator is also r,s,t-SPFN, and this is represented by Proof By applying r,s,t-SPFNs operating rules and subsequently the meaning of sin L , we can obtain the result in Eq. (12).
Proof Since L i =L ∀ i and therefore ϕ i L i = ϕ i L So, by n i=1 ϕ i = 1 and Eq. ( 12), we have ST-r,s,t- Then, relying on the precision of STF , we have Thus, we get the proof following Definition 3.
Proof Based on previous results, it can be easily verified.
Definition 13 A sine trigonometric r,s,t-spherical fuzzy ordered weighted geometric (ST-r,s,t-SPFOWG) operator is a mapping ST-r,s,t-SPFOWG: σ n → σ defined as where θ is the list of permutations.

Basic characteristics of the proposed AOs
This part explores the diverse connections among the proposed AOs along with key elements.
Theorem 7 From r,s,t-SPFNs L i , the operatives ST-r,s,t-SPFWG satisfy the inequalities and ST-r,s,t- www.nature.com/scientificreports/Proof For "n" r,s,t-SPFNs L i = (ρ i , χ i , ε i ) and normalized weight vector ϕ i > 0 , and taking and from Proposition 1, we have the option to obtain Hence, using Eqs.( 18) and ( 19), we derive the outcome.
Theorem 8 Let L i , L are r,s,t-spherical fuzzy numbers, then Proof Let L i , L are r,s,t-SPFNs, then by their operational laws, we get Utilizing the ST-r,s,t- SPFWA operator's monotonicity property, we achieve the required result.
Theorem 9 For r,s,t-SPFNs L i , L , we have Proof For r,s,t-SPFNs L i , L the result obtained using the operators of ST-r,s,t-SPFWA, ST-r,s,t-SPFWAG and sin L are again r,s,t-SPFNs.Thus, we derived results by applying Theorem 5.
Theorem 10 For r,s,t-SPFNs L i , L , and a real number ⋓ ∈ [0, 1] Proof Let L i , and L be any two r,s,t-SPFNs and ⋓ ∈ [0, 1] , where ⋓ be any real number.Subsequently, we possess and by Proposition 2, we get With the help of Eqs. ( 20), (21) and Definition 3, we get the result (1) which is valid for real ⋓ ∈ [0, 1] .
The MAGDM problem-solving process involves the following steps.
Step 1: Collect the preferences given by DnMs in the form of decision matrix D = I (h) ij including r,s,t-SPFS details.
Step 2: Build the normalized decision matrix R = p Step 3: Integrates DnMs values Step 4: Aggregate the overall rating values D = I ij of the alternative I i (i = 1, 2, ..., m) into the overall assess- ment value I i = (ρ i , χ i , ε i ) based on geometric r,s,t-SPF AO as defined in Definition 11.

Numerical analysis
This section offers a case study on assessing laptops to illustrate the utilization of the framework presented.It comprises four subsections: result analysis, sensitivity analysis, comparative study, and managerial implications.

Example:
A team of three DnMs specializing in laptops convened to conduct a comprehensive decision analysis involving multiple attributes aimed at selecting the most suitable laptop model.Each criterion represents a key aspect of the laptop's performance, while each attribute provides a finer-grained assessment of those attributes.In this context, the team has chosen the following four attributes: Vol:.( 1234567890) Size refers to the physical dimensions of the laptop, including factors such as thickness, weight, and overall form factor. Smaller and lighter laptops are often favored for their ease of transportation.Screen Quality ( C 3 ): The screen criterion assesses the visual display capabilities of the laptop, including factors such as resolution, color accuracy, and brightness.A high-quality screen enhances the user experience, particularly for tasks involving multimedia content.Sound Quality ( C 4 ): Sound quality pertains to the audio output capabilities of the laptop, including speaker performance and audio clarity.Good sound quality is essential for activities such as video conferencing, multimedia playback, and general entertainment.
For this decision problem, the DnMs have the weight vector ̟ = (0.3, 0.4, 0.3) T .Additionally, the weight vector assigned by the DnMs to the attributes is denoted as ϕ = (0.4,0.2, 0.1, 0.3) T , indicating the relative impor- tance of each criterion in the DM process.It's noteworthy that all attributes are considered benefit-type attributes.
Subsequently, the team applied the proposed r, s, t − SPF framework to identify the optimal model among the available options.This framework facilitates a systematic approach to MAGDM, enabling the team to effectively evaluate and compare the laptop models based on their performance across the established attributes.Through this detailed analysis, the team aims to make an informed decision that aligns with their preferences and requirements, ultimately selecting the laptop model that best fulfills their needs.
The calculation procedures are outlined as follows: Step 1: The r, s, t SPF data provided by the three DnMs is presented in Tables 1, 2 and 3, respectively.
Step 2: Since all four attributes are benefit types.Therefore, we do not need normalization.
Step 3: Utilizing the experts' weights, i.e., ̟ = (0.3, 0.4, 0.3) T and applying the ST-r,s,t-SPFWA operator, the collective data for each alternative is obtained and is shown in Table 4.
Step 6: Based on the above-derived score values, the ranking of alternatives is Vol.:(0123456789)

Influence of parameters
This section focuses on conducting a sensitivity analysis to assess how different parameters affect the ranking outcomes.
To showcase the reliability as well as uniformity of the illustration above, we check the sensitivity concerning different parameters such as r * , s * , and t * within a structured framework.For this, we set the values of s * = 4 and t * = 3 and explore different values for r * .By changing the value of r * , the ranking results of different choices stay consistent, i.e., I 4 > I 1 > I 2 > I 3 which shown in Table 5.Furthermore, upon fixing r * = 4, t * = 3 and increase the value of s * = 5, 7, 10, 13, 17, 20 in the proposed ST-r,s,t-SPFWG operator, it is noticeable that from Table 6 the ranking outcomes of choices remains unchanged i.e., I 4 > I 1 > I 2 > I 3 .Similarly, if we fix r * = 4, s * = 4 and vary the value of t * = 4, 7, 10, 13, 17, 20 in ST-r,s,t-SPFWG, again, it is noticeable, that from Table 7 analogous s * the ranking outcomes remains same.Therefore, the proposed approach exhibits isotonicity and stability under the ST-r,s,t-SPFWG operator across various values of r * , s * and t * .
Based on the data presented in this Table 8, it can be deduced that the optimal alternative identified by the proposed method aligns with the majority of existing approaches, thus affirming the validity of the proposed approach.We can further notice that ST-AOs of PyFS and SPFS cannot handle the data provided in the current problem.The analysis suggests that the existing AOs can be viewed as specific instances within the framework of the proposed method.Furthermore, this outcome indicates that the proposed method offers a broader approach than the existing AOs.
Based on the comparative analysis, the merits and outcomes of the reported framework are outlined as follows: (i).Compared to other assessment frameworks, the framed approach utilizes more reasonable input data, namely r, s, t-SPFNs, for evaluating alternatives.However, the data in the other assessment frameworks [28][29][30]35,36 doesn't take advantage of the three flexible parameters r, s, and t. The iclusion of these adjustable parameters broadens the scope of the application and allows for a more reasonable capture of data.(ii).Unlike the existing methods 18,19 , our proposed ST operators take into account the significance of trigonometry's characteristics, such as its periodicity and symmetry, in the analysis.This makes our approach superior to the existing r,s,t-SPF AOs.(iii).The framework presented requires only a few straightforward steps, highlighting its computational convenience.This accessibility makes it highly suitable for emergency decision support scenarios, where quick and effective DM is crucial.
The proposed study also exhibits certain drawbacks, which are enumerated as follows: I. One limitation of the outlined framework is that it requires prior knowledge of the weights for DnMs and attributes.Without this information upfront, the algorithm isn't applicable, which could be problematic for scenarios where these details aren't available beforehand.II.The developed ST AOs lack the capability to account for divisions among input arguments and may not be deemed valid in MAGDM problems, where attributes can be classified into distinct classes.

Managerial implications
The above analysis of laptop selection provides a comprehensive overview of the most commonly used technologies in the industry, highlighting their respective advantages and drawbacks.The proposed MAGDM framework effectively identifies the most pertinent laptop options, emphasizing those that are environmentally friendly, cost-effective, user-friendly, and capable of meeting substantial computing needs.This information offers valuable insights for both businesses and consumers, enabling them to make informed decisions based on their specific requirements.Additionally, policymakers can leverage this research to promote the adoption of laptops as essential tools for education, business, and personal use, particularly in underserved areas where access to traditional Table 8.Ranking order of the alternatives using different approaches.

Ranking values
Proposed approach I 4 > I 1 > I 2 > I 3 ST-q-ROFWA 30 I 2 > I 4 > I 1 > I 3 ST-q-ROFWG 30 ST-PFWA 28 Unable to aggregate ST-PFWG 28 Unable to aggregate ST-SPFWA 29 Unable to aggregate ST-SPFWG 29 Unable to aggregate power sources may be limited.The widespread adoption of laptops and the advancement of their technological recycling processes hold the potential to stimulate industrial growth and create employment opportunities in the technology sector.

Ethical approval
This material is the authors' own original work, which has not been previously published elsewhere.

Conclusion
The primary objective of this study was to introduce a novel perspective on operational laws and operators applicable to various r, s, t-SPFNs.We introduced STOLs and defined a new ST-r, s, t-SPFN to address this aim.Detailed discussions were conducted on the fundamental properties of these proposed laws.Additionally, we formulated several weighted averaging and geometric operators based on these laws to aggregate r, s, t-SPF information.The relationships between these operators were analyzed through derived inequalities, elucidating their correlations.The basic axioms of these operators were demonstrated to be satisfied within the proposed framework.Moreover, to tackle group DM problems, we developed a novel MAGDM which considers multiple decision-makers and alternatives within a r, s, t-SPF environment.The reliability and effectiveness of the developed algorithm were evaluated through a numerical example and compared with existing approaches.Through these analyses, it was observed that the presented algorithm and operators effectively manage a broader spectrum of information, rendering them highly capable of addressing DM Problems.
In the future, we aim to address the limitations of this study, as highlighted in the analysis section.To this end, we intend to integrate additional operators, such as the Maclaurin symmetric mean operator, the partition aggregation operator, and the power Muirhead mean operator, with the proposed ST AOs.Subsequently, we plan to develop an integrated weight calculation method by combining some subjective weighting methods, such as the level-based weight assessment method or the rank sum method, with an entropy-based approach.

Table 4 .
Aggregated values of experts by ST-r,s,t-SPFWA operator.

Table 5 .
Results with different value of r * .